A nonlinear lower bound on the practical combinational complexity

AbstractAn infinite sequence F = {fn}n = 1∞ of one-output Boolean functions with the following two properties is constructed: 1.(1)fn can be computed by a Boolean circuit with O(n) gates.2.(2)For any positive, nondecreasing, and unbounded function h : N → R, each Boolean circuit having an mh(m) separator requires a nonlinear number Ω(nh(n)) of gates to compute fn (e.g., each planar Boolean circuit requires Ω(n2) gates to compute fn).Thus, one can say that fn has linear combinational complexity and a nonlinear practical combinational complexity because the constant-degree parallel architectures used in practice have separators in O(mlog2 m)

Random walks on networks

Práce se zaobývá reverzibilními Markovovými řetězcemi, jejich reprezentací pomocí elektrických sítí, a metodami pro jejich studium převzanými z teorie elektrických sítí. Hlavním z prezentovaných výsledků je Pólyova věta, zaobývající sa přechodností náhod- ných procházek na celočíselných mřížkách. 1The thesis studies reversible Markov chains, their representation as electrical networks, and methods of analyzing them adapted from the theory of electrical networks. Main result presented is Pólya's theorem concerning random walks on integer lattices. 1Katedra pravděpodobnosti a matematické statistikyDepartment of Probability and Mathematical StatisticsFaculty of Mathematics and PhysicsMatematicko-fyzikální fakult

Random walks on networks

The thesis studies reversible Markov chains, their representation as electrical networks, and methods of analyzing them adapted from the theory of electrical networks. Main result presented is Pólya's theorem concerning random walks on integer lattices.